Paper 2, Section II, C

Differential Equations | Part IA, 2017

The current I(t)I(t) at time tt in an electrical circuit subject to an applied voltage V(t)V(t) obeys the equation

Ld2Idt2+RdIdt+1CI=dVdtL \frac{d^{2} I}{d t^{2}}+R \frac{d I}{d t}+\frac{1}{C} I=\frac{d V}{d t}

where R,LR, L and CC are the constant resistance, inductance and capacitance of the circuit with R0,L>0R \geqslant 0, L>0 and C>0C>0.

(a) In the case R=0R=0 and V(t)=0V(t)=0, show that there exist time-periodic solutions of frequency ω0\omega_{0}, which you should find.

(b) In the case V(t)=H(t)V(t)=H(t), the Heaviside function, calculate, subject to the condition

R2>4LCR^{2}>\frac{4 L}{C}

the current for t0t \geqslant 0, assuming it is zero for t<0t<0.

(c) If R>0R>0 and V(t)=sinω0tV(t)=\sin \omega_{0} t, where ω0\omega_{0} is as in part (a), show that there is a timeperiodic solution I0(t)I_{0}(t) of period T=2π/ω0T=2 \pi / \omega_{0} and calculate its maximum value IMI_{M}.

(i) Calculate the energy dissipated in each period, i.e., the quantity

D=0TRI0(t)2dtD=\int_{0}^{T} R I_{0}(t)^{2} d t

Show that the quantity defined by

Q=2πD×LIM22Q=\frac{2 \pi}{D} \times \frac{L I_{M}^{2}}{2}

satisfies Qω0RC=1Q \omega_{0} R C=1.

(ii) Write down explicitly the general solution I(t)I(t) for all R>0R>0, and discuss the relevance of I0(t)I_{0}(t) to the large time behaviour of I(t)I(t).

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